Skip to main content
SpringerLink
Log in

Parametric Characteristic Analysis for Generalized Frequency Response Functions of Nonlinear Systems

  • Published:
Circuits, Systems & Signal Processing Aims and scope Submit manuscript

Abstract

In order to explicitly reveal the relationship between system frequency response functions and model parameters which define system nonlinearities, and consequently unveil a direct connection from model parameters to system frequency response characteristics, a parametric characteristic analysis approach is proposed for Volterra systems described by a nonlinear differential equation (NDE). The parametric characteristics of the generalized frequency response functions (GFRFs) for the NDE model are established, and some important properties are discussed, which can explicitly reveal what model parameters contribute and how these parameters affect the GFRFs. Based on the parametric characteristic analysis, it is demonstrated how the system frequency domain characteristics are related to the system time domain model parameters and how the output frequency response function can now be determined explicitly with a detailed polynomial structure. These new results provide a significant and novel insight into the analysis and design of nonlinear systems in the frequency domain. Several examples are included to illustrate the results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Bedrosian, S.O. Rice, The output properties of Volterra systems driven by harmonic and Gaussian inputs. Proc. Inst. Electr. Electron. Eng. 59, 1688–1707 (1971)

    MathSciNet  Google Scholar 

  2. J.S. Bendat, Nonlinear System Analysis and Identification from Random Data (Wiley, New York, 1990)

    MATH  Google Scholar 

  3. S.A. Billings, Z.Q. Lang, Nonlinear systems in the frequency domain: Energy transfer filters. Int. J. Control 75(14), 1066–1081 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. S.A. Billings, J.C. Peyton-Jones, Mapping nonlinear integro-differential equations into the frequency domain. Int. J. Control 54, 863–879 (1990)

    Article  Google Scholar 

  5. S. Boyd, L.O. Chua, Fading memory and the problem of approximating nonlinear operators with Volterra series. IEEE Trans. Circuits Syst. CAS-32, 1150–1161 (1985)

    Article  MathSciNet  Google Scholar 

  6. D.A. George, Continuous nonlinear systems. Technical Report 355, MIT Research Laboratory of Electronics, Cambridge, MA, Jul. 24, 1959

  7. X.J. Jing, Z.Q. Lang, S.A. Billings, New bound characteristics of NARX model in the frequency domain. Int. J. Control 80(1), 140–149 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. X.J. Jing, Z.Q. Lang, S.A. Billings, G.R. Tomlinson, The parametric characteristic of frequency response functions for nonlinear systems. Int. J. Control 79(12), 1552–1564 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. X.J. Jing, Z.Q. Lang, S.A. Billings, Frequency domain analysis for suppression of output vibration from periodic disturbance using nonlinearities. J. Sound Vib. 314, 536–557 (2008)

    Article  Google Scholar 

  10. K.I. Kim, E.J. Powers, A digital method of modelling quadratic nonlinear systems with a general random input. IEEE Trans. Acoust. Speech Signal Process. 36, 1758–1769 (1988)

    Article  MATH  Google Scholar 

  11. S. Kotsios, Finite input/output representative of a class of Volterra polynomial systems. Automatica 33, 257–262 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Z.Q. Lang, S.A. Billings, Output frequency characteristics of nonlinear systems. Int. J. Control 64, 1049–1067 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Z.Q. Lang, S.A. Billings, Output frequencies of nonlinear systems. Int. J. Control 67, 713–730 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Z.Q. Lang, S.A. Billings, Energy transfer properties of nonlinear systems in the frequency domain. Int. J. Control 78, 345–362 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Z.Q. Lang, S.A. Billings, R. Yue, J. Li, Output frequency response functions of nonlinear Volterra systems. Automatica 43, 805–816 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. S.W. Nam, E.J. Powers, Application of higher-order spectral analysis to cubically nonlinear-system identification. IEEE Trans. Signal Process. 42(7), 1746–1765 (1994)

    Article  Google Scholar 

  17. P.W.J.M. Nuij, O.H. Bosgra, M. Steinbuch, Higher-order sinusoidal input describing functions for the analysis of non-linear systems with harmonic responses. Mech. Syst. Signal Process. 20, 1883–1904 (2006)

    Article  Google Scholar 

  18. P. Orlowski, Frequency domain analysis of uncertain time-varying discrete-time systems. Circuits Syst. Signal Process. 26(3), 293–310 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. R.K. Pearson, Discrete Time Dynamic Models (Oxford University Press, London, 1994)

    Google Scholar 

  20. J.C. Peyton Jones, Automatic computation of polyharmonic balance equations for non-linear differential systems. Int. J. Control 76(4), 355–365 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. W.J. Rugh, Nonlinear System Theory—The Volterra/Wiener Approach (Johns Hopkins University Press, Baltimore, 1981)

    MATH  Google Scholar 

  22. I.W. Sandberg, Expansions for nonlinear systems. Bell Syst. Tech. J. 61(2), 159–199 (1982)

    MathSciNet  MATH  Google Scholar 

  23. S. Sastry, Nonlinear Systems: Analysis, Stability, and Control (Springer, New York, 1999)

    MATH  Google Scholar 

  24. M.A. Shah, M.A. Franchek, Frequency-based controller design for a class of nonlinear systems. Int. J. Robust Nonlinear Control 9(12), 825–840 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Solomou, C. Evans, D. Rees, N. Chiras, Frequency domain analysis of nonlinear systems driven by multiharmonic signals. In Proceedings of the 19th IEEE Conference on Instrumentation and Measurement Technology, vol. 1, pp. 799–804 (2002)

  26. M. Solomou, D. Rees, N. Chiras, Frequency domain analysis of nonlinear systems driven by multiharmonic signals. IEEE Trans. Instrum. Meas. 53(2), 243–250 (2004)

    Article  Google Scholar 

  27. W. Van Moer, Y. Rolain, A. Geens, Measurement-based nonlinear modeling of spectral regrowth. IEEE Trans. Instrum. Meas. 50(6), 1711–1716 (2001)

    Article  Google Scholar 

  28. V. Volterra, Theory of Functionals of Integral and Integrodifferential Equations (Dover, New York, 1959)

    Google Scholar 

  29. R. Yue, S.A. Billings, Z.-Q. Lang, An investigation into the characteristics of non-linear frequency response functions. Part 1: Understanding the higher dimensional frequency spaces. Int. J. Control 78(13), 1031–1044 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Sound and Vibration Research, University of Southampton, Southampton, SO15 5DD, UK

    Xing Jian Jing

  2. Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK

    Xing Jian Jing, Zi Qiang Lang & Stephen A. Billings

Authors
  1. Xing Jian Jing
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zi Qiang Lang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stephen A. Billings
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xing Jian Jing.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jing, X.J., Lang, Z.Q. & Billings, S.A. Parametric Characteristic Analysis for Generalized Frequency Response Functions of Nonlinear Systems. Circuits Syst Signal Process 28, 699–733 (2009). https://doi.org/10.1007/s00034-009-9106-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-009-9106-7

Keywords

Search

Navigation